Gradient help

  • Thread starter Juntao
  • Start date
45
0
Consider the function z=f(x,y). If you start at the point (4,5) and move toward the point (5,6), the direction derivative is sqrt(2). Starting at (4,5) and moving toward (6,6), the directional derivative is sqrt(5). Find gradient f at (4,5).

Okay, this is probably a simple problem, but I don't know how to start it. Help appreciated.
 

Hurkyl

Staff Emeritus
Science Advisor
Gold Member
14,845
17
Do you know a formula that relates gradients and directional derivatives?

What do you get when you plug what you know into that formula? (represent unknowns with variables)
 
45
0
Forumula...*scratches head*

Um, I think this is the one, hopefully:
The derivative of f at Po in the direction of u= gradient f dot with u = magnitude grad f * magnitude u * cos theta = magnitude grad f * cos theta.

Or like D_u f= grad f (dot) u = |grad f| |u| cos theta = |grad f| cos theta

Maybe that'll help me out?
 

HallsofIvy

Science Advisor
Homework Helper
41,712
876
D_u f= grad f (dot) u
Yes, that's true as long as u is a unit vector (has length 1).

Another way of saying this is that the directional derivative is
f_x u_x+ f_y u_y where f_x is the partial derivative and u_x is the x component of unit vector u.

Find unit vectors in both the given directions and write out
f_x u_x+ f_y u_y= √(2) and
f_x v_x+ f_y v_y= √(5).

That gives you two equations for f_x and f_y.
 

Hurkyl

Staff Emeritus
Science Advisor
Gold Member
14,845
17
D_u f= grad f (dot) u
That's the one of which I was thinking.

You know two instances of D_u f and u. (don't forget that u is a unit vector!), so grad f is the unknown. You have two equations, so you should be able to solve for the two components of grad f.
 

Related Threads for: Gradient help

Replies
3
Views
1K
Replies
8
Views
2K
Replies
3
Views
2K
  • Posted
Replies
5
Views
2K
  • Posted
Replies
1
Views
2K
  • Posted
Replies
2
Views
2K
  • Posted
Replies
1
Views
1K
  • Posted
Replies
8
Views
4K

Physics Forums Values

We Value Quality
• Topics based on mainstream science
• Proper English grammar and spelling
We Value Civility
• Positive and compassionate attitudes
• Patience while debating
We Value Productivity
• Disciplined to remain on-topic
• Recognition of own weaknesses
• Solo and co-op problem solving

Hot Threads

Top